What is the integral of (x-1)/((x+2)^2) ?

The integral of (x-1)/((x+2)^2) is ln|x+2|+3/(x+2)+C

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integral of (x-1)/((x+2)^2)

\int \frac{x - 1}{( x + 2 ) ^{2}} d x

ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

Solution steps

\int \frac{x - 1}{( x + 2 ) ^{2}} d x

Expand

\frac{x - 1}{( x + 2 ) ^{2}} : \frac{x}{( x + 2 ) ^{2}} - \frac{1}{( x + 2 ) ^{2}}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{( x + 2 ) ^{2}} d x - \int \frac{1}{( x + 2 ) ^{2}} d x

\int \frac{x}{( x + 2 ) ^{2}} d x = ln ∣ x + 2 ∣ + \frac{2}{x + 2}

\int \frac{1}{( x + 2 ) ^{2}} d x = - \frac{1}{x + 2}

= ln ∣ x + 2 ∣ + \frac{2}{x + 2} - (- \frac{1}{x + 2})

Simplify

ln ∣ x + 2 ∣ + \frac{2}{x + 2} - (- \frac{1}{x + 2}) : ln ∣ x + 2 ∣ + \frac{3}{x + 2}

= ln ∣ x + 2 ∣ + \frac{3}{x + 2}

Add a constant to the solution

= ln ∣ x + 2 ∣ + \frac{3}{x + 2} + C

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What is the integral of (x-1)/((x+2)^2) ?
The integral of (x-1)/((x+2)^2) is ln|x+2|+3/(x+2)+C